Computation of Periodic Solutions of Conservative Systems with Application to the 3-Body Problem

Abstract

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simó, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

“…These algorithms have been extensively used for computing the forced response and limit cycles of nonlinear dynamical systems [6][7][8][9][10][11]. Doedel and co-workers used them for the computation of periodic orbits during the free response of conservative systems [12,13].…”

Computation of low energy Earth-to-Moon transfers with moderate flight time

“…We first note that there exists a family of unstable periodic orbits around the Lagrange point L 1 as well as L 2 and L 3 [13,18]. See also [10] for numerical computations of the periodic orbits emanating from L 1 by AUTO. The periodic orbits have stable and unstable manifolds and their transverse intersection yields chaotic dynamics [1,20].…”

Sun-perturbed earth-to-moon transfers with low energy and moderate flight time

“…(see Szebehely 1967;Koon et al, 2000a). Numerical computations of the periodic orbits emanating from L 1 by AUTO were also given in Doedel et al (2003). The periodic orbits have stable and unstable manifolds and their transverse intersection yields chaotic dynamics (Wiggins, 1990;Alligood et al, 1996).…”

“…See K. Yagasaki (in preparation) for more details on the incorporation of DOP853 in Dynamics. The periodic orbit was numerically computed by AUTO as in Doedel et al (2003).…”